Question:

Please convert the below content to rendered equations without any changes and errors.We solve the **overall transfer function ( \frac{C(s)}{R(s)} )** using **block diagram reduction rules** (series, feedback, and summing rules). ([Fiveable][1]) --- # Step 1: Identify blocks From the diagram: Forward path blocks: [ 10 \rightarrow \frac{20}{s+5} \rightarrow \frac{100}{s^2} ] Feedback blocks: Top feedback: [ \frac{5}{s} ] Bottom feedback: [ \frac{2}{s} ] --- # Step 2: Combine series blocks in forward path First combine: [ \frac{20}{s+5} \times \frac{100}{s^2} ] [ = \frac{2000}{s^2(s+5)} ] Now include the gain **10** [ G(s)=10 \times \frac{2000}{s^2(s+5)} ] [ G(s)=\frac{20000}{s^2(s+5)} ] --- # Step 3: Inner feedback loop The feedback around the second summing junction is [ H_1(s)=\frac{5}{s} ] Closed-loop formula: [ G_{eq}=\frac{G}{1+GH} ] Substitute [ G_{eq}=\frac{\frac{20000}{s^2(s+5)}}{1+\frac{20000}{s^2(s+5)}\frac{5}{s}} ] Multiply: [ \frac{20000}{s^2(s+5)}\times\frac{5}{s} ] [ =\frac{100000}{s^3(s+5)} ] Thus [ G_{eq}=\frac{20000}{s^2(s+5)+100000} ] --- # Step 4: Outer feedback loop Outer feedback block [ H_2(s)=\frac{2}{s} ] Closed-loop formula again: [ T(s)=\frac{G_{eq}}{1+G_{eq}H_2} ] Substitute [ T(s)=\frac{\frac{20000}{s^2(s+5)+100000}} {1+\frac{20000}{s^2(s+5)+100000}\frac{2}{s}} ] Multiply: [ \frac{20000}{s^2(s+5)+100000}\times\frac{2}{s} ] [ =\frac{40000}{s[s^2(s+5)+100000]} ] --- # Step 5: Final transfer function After simplifying, [ \boxed{ \frac{C(s)}{R(s)}= \frac{20000s} {s^4+5s^3+100000s+40000} } ] --- # ✅ Final Answer [ \boxed{ \frac{C(s)}{R(s)}= \frac{20000s}{s^4+5s^3+100000s+40000} } ] --- ✅ If you want, I can also show a **much easier 3-step shortcut (using Mason’s Gain Formula)** that professors often expect in exams for this type of block diagram. [1]: https://fiveable.me/introduction-dynamic-systems/unit-6/block-diagram-reduction-techniques/study-guide/YdvsMoZCYXxNe5dS?utm_source=chatgpt.com "Block Diagram Reduction Techniques | Intro to Dynamic Systems Class Notes"